The first Darboux problem for wave equations with nonlinear dissipative term is considered. The uniqueness, local and global existence and blow-up of solutions of the problem mentioned are investigated. The paper’s originality is the coalescence of two standard methods: a priori estimate of solutions in the class of continuous functions is given by energetic methods; basing on this result a priori estimate in the class of continuously differentiable functions using classical method of characteristics is obtained.